The thermodynamic description of mixtures

We now leave pure materials and the limited but important changes they can undergo and examine mixtures. We shall consider only homogeneous mixtures, or solutions, in which the composition is uniform however small the sample. The component in smaller abundance is called the solute and that in larger abimdance is the solvent. These terms, however, are normally but not invariably reserved for solids dissolved in Kquids one liquid mixed with another is normally called simply a mixture of the two liquids. In this chapter we consider mainly nonelectrolyte solutions, where the solute is not present as ions. Examples are sucrose dissolved in water, sulfur dissolved in carbon disulfide, and a mixture of ethanol and water. Although we also consider some of the special problems of electrolyte solutions, in which the solute consists of ions that interact strongly with one another, we defer a full study until Chapter 5. The measures of concentration commonly encoimtered in physical chemistry are reviewed in Further information 3.2. 

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The objective of this chapter and the two that follow is to illustrate how the principles introduced in Chapter 8 for the thermodynamic description of mixtures together with the calculational procedures of Chapter 9 can be used to study many different types of phase equilibria important in chemical engineering practice. In particular, the following are considered ... 

The thermodynamic description of the formation of mlcroporous systems by means of the phase diagrams, eis illustrated in Figures 1 to 3, is based on the assumption of thermodynamic equilibrium. It predicts under what conditions of temperature and composition a system will separate into two phases and the ratio of the two phases in the heterogeneous mixture. As related to the membrane formation procedure, the thermodynamic description predicts the overall porosity that will be obtained at specified states. However, no information is provided about the pore sizes, which are determined by the spatial distribution of the two phases. Equilibrium thermodynamics is not able to offer any explanation about structural variations within the membrane cross-section that is, whether the membrane has a symmetric or asymmetric structure or a dense skin at the surface. These... 

Thus, the interrelationships provided by Eqs. 8.2-8 through 8.2-15 are really restrictions on the mixture equation of state. As such, these equations are important in minimizing the amount of experimental data necessary in evaluating the thermodynamic, properties of mixtures, in simplifying the description of multicomponent systems, and in testing the consistency of certain types of experimental data (see Chapter 10). Later in this chapter we show how the equations of change for mixtures and the Gibbs-Duhem equations provide a basis for the experimental determination of partial molar properties. 

The difficulty with this description is that Ga and Gb are not separately measurable, because, as a result of Eq. 9.10-3, it is not possible to vary the number of moles of cations holding the number of moles of anions fixed, or vice versa. (Even in mixed electrolyte solutions, that is, solutions of several electrolytes, the condition of overall electrical neutrality makes it impossible to vary the number of only one ionic species.) To maintain the present thermodynamic description of mixtures and, in particular, the concept of the partial molar Gibbs energy, we instead consider a single electrolyte solu-,tion to be a three-component system solvent, undissociated electrolyte, and dissociated electrolyte. Letting Nab,d be the moles of dissociated electrolyte, we then have... 

The theme of this chapter is that, while thermodynamic descriptions of mixtures involve a large number of equations, those equations tend to fall into a few repeated patterns. By recognizing the patterns, we not only broaden our understanding, but we also reduce the number of different things that must be mastered. 

Reich and Cohen ° were the first who proposed the thermodynamic description of the effect of interface on the phase separation in pol5mier mixtures. They considered the lattice-cell model, where the coordination number of lattice points is uniform throughout the lattice, but, in the surface layer, it should be lower than in bulk. The number of pair interactions in the total sample of N lattice sites is... 

So, one of the main limitations for a valuable description of this kind of mixtures is the persistent lack of knowledge in the thermodynamic description of polyelectrolytes in solution. However, recently, new expressions of the free energy of a polyelectrolyte chain were proposed by Leibler, Pezron et al. [44]. More generally, a microphase separation for polyelectrolytes for which the macromolecular backbone is not soluble in water were proposed independently by Borue et al. [45] and Leibler et al. [46]. An approach, taking into account the role of polyelectrolyte counterions was also proposed [47]. 

Von Neumann18 did more than coin these words. He showed how the statistical matrix can be generalized to include the description of mixtures, and he succeeded, mainly by this device, in laying the foundation for the quantum mechanical counterpart of thermodynamics. 

Chapters 7 to 9 apply the thermodynamic relationships to mixtures, to phase equilibria, and to chemical equilibrium. In Chapter 7, both nonelectrolyte and electrolyte solutions are described, including the properties of ideal mixtures. The Debye-Hiickel theory is developed and applied to the electrolyte solutions. Thermal properties and osmotic pressure are also described. In Chapter 8, the principles of phase equilibria of pure substances and of mixtures are presented. The phase rule, Clapeyron equation, and phase diagrams are used extensively in the description of representative systems. Chapter 9 uses thermodynamics to describe chemical equilibrium. The equilibrium constant and its relationship to pressure, temperature, and activity is developed, as are the basic equations that apply to electrochemical cells. Examples are given that demonstrate the use of thermodynamics in predicting equilibrium conditions and cell voltages. 

It has been said chemists have solutions 3 Solutions are involved in so many chemical processes1 that we must have the mathematical tools to comfortably work with them, and thermodynamics provides many of these tools. Thermodynamic properties such as the chemical potential, partial molar properties, fugacities, and activities, provide the keys to unlock the description of mixtures. 

The formal description of thermodiffusion in the critical region has been discussed in detail by Luettmer-Strathmann [79], The diffusion coefficient of a critical mixture in the long wavelength limit contains a mobility factor, the Onsager coefficient a = ab + Aa, and a thermodynamic contribution, the static structure factor S(0) [7, 79] ... 

Activity as a function was introduced by Lewis in 1908, and a full description was given by Lewis and Randall [74] in 1923. The activity a of a substance i can be defined [75.76] as a value corresponding to the mole fraction of the substance i in the given phase. This value is in agreement with the thermodynamic potential of the ideal mixture and gives the real value of this potential. 

In Chap. 6 we treated the thermodynamic properties of constant-composition fluids. However, many applications of chemical-engineering thermodynamics are to systems wherein multicomponent mixtures of gases or liquids undergo composition changes as the result of mixing or separation processes, the transfer of species from one phase to another, or chemical reaction. The properties of such systems depend on composition as well as on temperature and pressure. Our first task in this chapter is therefore to develop a fundamental property relation for homogeneous fluid mixtures of variable composition. We then derive equations applicable to mixtures of ideal gases and ideal solutions. Finally, we treat in detail a particularly simple description of multicomponent vapor/liquid equilibrium known as Raoult s law. 

For a pure supercritical fluid, the relationships between pressure, temperature and density are easily estimated (except very near the critical point) with reasonable precision from equations of state and conform quite closely to that given in Figure 1. The phase behavior of binary fluid systems is highly varied and much more complex than in single-component systems and has been well-described for selected binary systems (see, for example, reference 13 and references therein). A detailed discussion of the different types of binary fluid mixtures and the phase behavior of these systems can be found elsewhere (X2). Cubic ecjuations of state have been used successfully to describe the properties and phase behavior of multicomponent systems, particularly fot hydrocarbon mixtures (14.) The use of conventional ecjuations of state to describe properties of surfactant-supercritical fluid mixtures is not appropriate since they do not account for the formation of aggregates (the micellar pseudophase) or their solubilization in a supercritical fluid phase. A complete thermodynamic description of micelle and microemulsion formation in liquids remains a challenging problem, and no attempts have been made to extend these models to supercritical fluid phases. 

In this chapter we are concerned with developing the equations of energy conservation to be used in the thermodynamic analysis of systems of pure substances. (The thermodynamics of mixtures is more complicated and will be considered in later chapters.) To emphasize both the generality of these equations and the lack of detail necessary, we write these energy balance equations for a general black-box system. For contrast, and also because a more detailed description will be.useful in Chapter 4, the rudiments of the more detailed microscopic description are provided in the final, optional section of this chapter. This microscopic description is not central to our development of thermodynamic principles, is suitable only for advanced students, and may be omitted. 

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